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Q.
The arithmetic mean of two numbers is $18\frac{3}{4}$ and the positive square root of their product is $15.$ The larger of the two numbers is
NTA AbhyasNTA Abhyas 2022
Solution:
Let $a$ and $b$ be the positive numbers.
Then $\frac{a + b}{2}=18\frac{3}{4}\Rightarrow a+b=\frac{75}{2}$
and $\sqrt{a b}=15\Rightarrow ab=225$
so, $\left(a - b\right)^{2}=\left(a + b\right)^{2}-4ab=\left(\frac{75}{2}\right)^{2}-4\times 225$
$=\left(\frac{75}{2} + 30\right)\left(\frac{75}{2} - 30\right)=\frac{135}{2}\times \frac{15}{2}$
$\Rightarrow a-b=\pm\left(\frac{15}{2} \times 3\right)=\pm\left(\frac{45}{2}\right)$
Case1 $ \rightarrow a+b=\frac{75}{2},a-b=\frac{45}{2}$
$\Rightarrow a=30,b=\frac{45}{2}$
Case2 $ \rightarrow a+b=\frac{75}{2},a-b=-\frac{45}{2}$
$\Rightarrow a=\frac{15}{2},b=30$
So, larger of the two numbers is $30$